Realization we are moving in a system

Hello, I am having one question about whether we can realize we are moving in a system, or we think that everything around us is moving away from us.
This question popped up while I was thinking about two let's say planets, orbiting each other... If we are to live on one of them, we'd probably be able to see that the other is moving around us... and doing the calculations in relative coordinates we could understand we are also moving around our CM...
However, what about the motion of the CM? In what way could we "realize" that we are not just orbiting each other, but we are also moving through the whole space?

If there is no third body with which to compare, there is no good way to define the "motion" of the center of mass. As far as the 2 bodies are concerned, the center of mass is stationary. If there is a third body (I'm using this third body to mean an inertial reference frame), we can talk about motion of the center of mass of the 2 bodies with respect to the third one. But if no such third body exists, there's only arbitrary ways to define the CoM motion by arbitrary construction of arbitrary inertial reference frames.